The main divergence here from traditional stochastic modeling of assets prices is that, in limit order books, the price is viewed as a discrete (not continuous) process. We don't assume that the price gradually changes, but rather jumps from one price to another on the grid separated by multiples of the defined tick size. In fact, in the limit order book, there is no single price of an asset, but instead there are only orders. For this purpose the mid-price is often modeled as an indication of the current price[1]}. This too is a discrete process since the mid-price is just the average of the best bid and best ask. In this way there is a fundamental difference to understanding the price process via limit order books as opposed to more traditional methods, such as Itô processes[2]}, [3]}.
